!0 Cubic Function Notes For Sam

March 24, 2008

Cubic Functions Vocabulary Cubic Function: a polynomial function of degree 3. Zero: For a polynomial function f, each solution of the equation f(x) = 0 Story Problem Example: Intro Scenario: A space shuttle travels at a speed of S(t) = 0.000559t^3 + 0.0303t^2 + 13.6t Travels at above speed until the booster rockets fall off at height of 3000 miles. For the answer, plug the equation into the calculator and find intersection so: Y= 0.000559x^3 + 0.0303x^2 + 13.6x Y= 3000 Set your minimums and maximums and the picture should look like…

The answer is 118.917

Cubic – Standard Form Cubic form equation: f(x) = ax³ + bx² + cx + d Intercept form of Cubic form equation: f(x) = a(x-p) (x-q) (x-r) - p, q, r are the intercepts

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So in a graph p, q, and r would be (graph to the left)

From Intercepts to Standard Cubic Equation You want to get f(x) = ax³ + bx² + cx + d out of f(x) = a(x-p) (x-q) (x-r). Example: Intercepts given are -5, -3, 0. The equations passes through the points (1, -6) Plug in points into the intercept form f(x) = a (x-p) (x-q) (x-r) f(x) = a (x + 5) (x + 3) (x - 0) Since x – o = x, we can move x to the front. We end up with… f(x) = ax(x + 5) (x + 3) To find the missing “a”… Add in the given coordinates.. -6 = a (1) (1 + 5) (1 + 3) -6 = a (1) (6) (4) -6 = 24a

a = -¼ To write the final equation, add in all the components…

f(x) = -¼x (x + 5) (x + 3) f(x) = -¼x (x + 5) (x + 3) is the final answer From Intercepts to Graphing -

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-5, -3 are the intercepts - -¼ is the steepness of the line f(x) = -¼x (x + 5) (x + 3) should look like

Standard From and Factoring (VERY IMPORTANT) From f(x) = (x + p) (x + q) (x + r) to x³ + x² + x = 0 .. etc. Factor Theorem: States let f(x) be a polynomial. Then x – k is a fact iff (if and only if) f(k) = 0. g(x) = x³ - 7x² + 4x + 12 Step 1: Find the factors of 12. In this case, the factors would be +/- 1 , +/- 2, +/- 3, +/- 4, +/- 6 Step 2: Use synthetic substitution for all of the factors until the remainder is zero. (guess and check)